Remark
Tag 0CXM, permanent. Cite it as [Bourbaki, Tag 0CXM].
Commutative Algebra, chapter IX, § 3. Corps de représentants
\* The preceding proof applies more generally to the case where one assumes only that A is a henselian local ring*. The proof of uniqueness uses the hypothesis that the local ring A is separated, but not that it is complete.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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